Actions of compact quantum groups
These lecture notes deal with aspects of the theory of actions of compact quantum groups on C$^*$-algebras (locally compact quantum spaces). After going over the basic notions of isotypical components and reduced and universal completions, we look at crossed and smash product C$^*$-algebras, up to the statement of the Takesaki–Takai–Baaj–Skandalis duality. We then look at two special types of actions, namely homogeneous actions and free actions. We study the actions which combine both types, the quantum torsors, and show that more generally any homogeneous action can be completed to a free action with a discrete, classical set of ‘quantum orbits’. We end with a combinatorial description of the homogeneous actions for the free orthogonal quantum groups.